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Solids of Revolution: Volume
Review: The Disk Method
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The Disk Method
If a region in the plane is revolved about a line, the resulting
solid is a solid of revolution, and the line is called the axis
of revolution.
The simplest such solid is a right
circular cylinder or disk, which is
formed by revolving a rectangle
about an axis adjacent to one
side of the rectangle,
as shown in Figure 7.13.
Figure 7.13
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The Disk Method
Divide the region into n equal disks
Find the volume of each disk, to get an approximate
volume.
As the number of disks approaches infinity, we
approach the actual volume:
ANIMATION
Figure 7.13
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5.
The Washer Method
The disk method can be extended to cover solids of
revolution with holes by replacing the representative disk
with a representative washer.
The washer is formed by revolving
a rectangle about an axis,
as shown in Figure 7.18.
If r and R are the inner and outer radii
of the washer and w is the width of the
washer, the volume is given by
Volume of washer = π(R2 – r2)w.
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Figure 7.18
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The Washer Method
To see how this concept can be used to find the volume of
a solid of revolution, consider a region bounded by an
outer radius R(x) and an inner radius r(x), as shown in
Figure 7.19.
Figure 7.19
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7.
Example 3 – Using the Washer Method
Find the volume of the solid formed by revolving the region
bounded by the graphs of
about the
x-axis, as shown in Figure 7.20.
Figure 7.20
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The Washer Method
If the region is revolved about its axis of revolution, the
volume of the resulting solid is given by
Note that the integral involving the inner radius represents
the volume of the hole and is subtracted from the integral
involving the outer radius.
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Example 3 – Solution
In Figure 7.20, you can see that the outer and inner radii
are as follows.
Integrating between 0 and 1 produces
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Practice
Find the volume of the solid formed by revolving
the region bounded by the graphs of f(x)= - x^2 +5x+3
and g(x) = -x + 8 about the x-axis.
SETUP the integral and then finish with your calculator.
Approximately 442.34 cubic units
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12.
The Washer Method
So far, the axis of revolution has been horizontal and you
have integrated with respect to x. In the Example 4, the
axis of revolution is vertical and you integrate with respect
to y. In this example, you need two separate integrals to
compute the volume.
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13.
Example 4 – Integrating with Respect to y, Two-Integral Case
Find the volume of the solid formed by revolving the region
bounded by the graphs of y = x2 + 1, y = 0, x = 0, and x = 1
about y-axis, as shown in Figure 7.21.
Figure 7.21
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Example 4 – Solution
For the region shown in Figure 7.21, the outer radius is
simply R = 1.
There is, however, no convenient formula that represents
the inner radius.
When 0 ≤ y ≤ 1, r = 0, but when 1 ≤ y ≤ 2, r is determined
by the equation y = x2 + 1, which implies that
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Example 4 – Solution
cont’d
Using this definition of the inner radius, you can use two
integrals to find the volume.
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Example 4 – Solution
Note that the first integral
cont’d
represents the volume
of a right circular cylinder of radius 1 and height 1.
This portion of the volume could have been determined
without using calculus.
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18.
Solids with Known Cross Sections
With the disk method, you can find the volume of a solid
having a circular cross section whose area is A = πR2.
This method can be generalized to solids of any shape, as
long as you know a formula for the area of an arbitrary
cross section.
Some common cross sections are squares, rectangles,
triangles, semicircles, and trapezoids.
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19.
Solids with Known Cross Sections
Figure 7.24
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Example 6 – Triangular Cross Sections
Find the volume of the solid shown in Figure 7.25.
The base of the solid is the region bounded by the lines
and x = 0.
Figure 7.25
The cross sections perpendicular to the x-axis are
equilateral triangles.
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Example 6 – Solution
The base and area of each triangular cross section are as
follows.
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Example 6 – Solution
cont’d
Because x ranges from 0 to 2, the volume of the solid is
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